American Mathematical Monthly Contents March 2011

ARTICLES

Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2011 to Joseph A. Gallian for Distinguished Service to Mathematics

Was Cantor Surprised?

Fernando Q. Gouvêa
We look at the circumstances and context of Cantor’s famous remark, “I see it, but
I don’t believe it.” We argue that, rather than denoting astonishment at his result, the remark
pointed to Cantor’s worry about the correctness of his proof. (Free sample pdf)

On Legendre’s Work on the Law of Quadratic Reciprocity

Steven H.Weintraub
Legendre was the first to state the law of quadratic reciprocity in the form in which we know it and he was able to prove it in some but not all cases, with the first complete proof being given by Gauss. In this paper we trace the evolution of Legendre’s work on quadratic reciprocity in his four great works on number theory.

Equimodular Polynomials and the Tritangency Theorems of Euler, Feuerbach, and Guinand

Alexander Ryba and Joseph Stern
We strengthen a result of Lehmer, obtaining a new necessary condition for the roots of a complex polynomial to have equal modulus. From this we derive the famous theorem of Feuerbach, as well as the less well-known theorems of Euler and Guinand on the tritangent centers of a triangle. The latter theorems constrain the possible locations of the incenter and excenters subject to fixed locations for the circumcenter and orthocenter.

The Sharkovsky Theorem: A Natural Direct Proof

Keith Burns and Boris Hasselblatt
We give a natural and direct proof of a famous result by Sharkovsky that gives a complete description of possible sets of periods for interval maps. The new ingredient is the use of Štefan sequences.

A First Look at Differential Algebra

John H. Hubbard and Benjamin E. Lundell
This article is an introduction to the common algebraic methods used to study both solutions to polynomial equations and solutions to differential equations: Galois theory and differential Galois theory. We develop both theories simultaneously by studying the solutions to the polynomial equation x5- 4x2 - 2 = 0 and the solutions to the differential equation u' = t - u2.

NOTES

Lines of Best Fit for the Zeros and for the Critical Points of a Polynomial

Grant Keady
Combining results presented in two papers in this MONTHLY yields the following elementary result. Any line of best fit for the zeros of a polynomial is a line of best fit for its critical points.

Regular Matchstick Graphs

Sascha Kurz and Rom Pinchasi
A matchstick graph is a plane geometric graph in which every edge has length 1 and no two edges cross each other. It was conjectured that no 5-regular matchstick graph exists. In this paper we prove this conjecture.

A Recursive Scheme for Improving the Original Rate of Convergence to the Euler–Mascheroni Constant

Edward Chlebus
We have used Euler–Maclaurin summation to develop a recursive scheme for modifying the original approximation for the Euler–Mascheroni constant γ. Convergence to γ resulting from successively employing the proposed scheme has been significantly accelerated while the form of the approximation originally introduced by Euler is still preserved.

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